After a raise I assaign my value range as broadway cards that did not hit.
In general stats show that villain is not raising or check raising bets.
I think in this case that was the fact I did not consider enough and did not pot control to a minimum due to pocket pairs he might hold.

Turn:($2.18, 2 players) 2Hero bets $1.10, BTN calls $1.10

Here I go for a bet to make him fold some brodways. I make only a half a pot to keep all flush in too.

River:($4.38, 2 players) 2Hero bets $3.50, BTN calls $3.50

River bet I suppose is really bad because there is no value if i bet so big.
Basic idea here would be to make him fold all flus cards and all broadways that are left.

We could think of a donk bet here on the flop, since we would love to pick up the pot here, plus if we get called we have a lot of equity.
If you think BU is cbetting a lot here on this flop you can go for a check/raise here. I don't really like the idea of check/calling and donking on this turn.

As played, betting the river is bad.
We don't get called by worse and we are trying to make fold hands that we do beat and are probably not betting.